We study the geometry of linear networks with one-dimensional convolutional layers. The function spaces of these networks can be identified with semi-algebraic families of polynomials admitting sparse factorizations. We analyze the impact of the network's architecture on the function space's dimension, boundary, and singular points. We also describe the critical points of the network's parameterization map. Furthermore, we study the optimization problem of training a network with the squared error loss. We prove that for architectures where all strides are larger than one and generic data, the non-zero critical points of that optimization problem are smooth interior points of the function space. This property is known to be false for dense linear networks and linear convolutional networks with stride one.

翻译：本文研究具有一维卷积层的线性网络的几何结构。这些网络的函数空间可视为具有稀疏分解的半代数多项式族。我们分析了网络架构对函数空间的维数、边界和奇点的影响，并描述了网络参数化映射的临界点。进一步，我们研究了采用平方误差损失训练网络的优化问题。我们证明：对于所有步长均大于1的架构及一般数据，该优化问题的非零临界点是函数空间的光滑内点。已知这一性质对于稠密线性网络和步长为1的线性卷积网络并不成立。